The 1 meter length is unimportant.
I assume we're making two cuts uniformly distributed along the bar.
So let and be iid U(0,1) rvs.
I checked the three case and each case we want the largest cut to be less than .5.
The cuts are , and ,
BUT here I'm assuming which is really the order stats.
I'm trying to figure the logic here.
We need
The joint density of these two is